The Dynamical History of the Kepler-221 Planet System

Antoine C. Petit; Chris W. Ormel; Shuo Huang; Tian Yi

REVIEW 3 major objections 5 minor 1 cited by

The Dynamical History of the Kepler-221 Planet System

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Kepler-221's odd resonance chain is the fossil of a five-planet system in which two planets merged, and a four-phase simulation can reproduce the observed arrangement.

desk verdict A solid, inventive four-phase merger scenario for Kepler-221 that advances the resonant-chain story, with a real age-vs-tidal-strength tension the authors acknowledge but do not resolve. read the letter →

arxiv 2502.01736 v2 pith:I5ZXC3XO submitted 2025-02-03 astro-ph.EP

Tian Yi , Chris W. Ormel , Shuo Huang , Antoine C. Petit This is my paper · ORCID

classification astro-ph.EP

keywords Kepler-221three-bodyresonanceplanet-planetmergerchaininstabilitytidaldissipationorbitalmigrationexoplanetarchitecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the Kepler-221 system did not always look like it does today. It argues that the present architecture, with planets b, c, and e in a 6:3:1 three-body resonance while the intermediate planet d stays out of it, is the end product of a four-phase history: five planets formed in a chain of first-order resonances, the chain broke after disk dispersal, two planets merged to become d, and tidal dissipation later rebuilt and stretched the (b, c, e) resonance to the observed period ratios. The argument matters because Kepler-221 is a case where simple disk migration cannot explain what is seen, and because the proposed history yields concrete, testable predictions about the planets' masses and tidal properties. A full N-body simulation is shown to pass through all four phases and arrive at the observed period ratios, with the key quantitative condition being a slope $a_{cde} > 1.56$.

What carries the argument

The load-bearing objects are the zeroth-order three-body resonance, an angle built from the mean longitudes of b, c, and e, $\phi_{3BR}=2\lambda_b-5\lambda_c+3\lambda_e$, whose libration defines the 6:3:1 chain, and the expansion slope $a_{cde}$, the ratio in which the period ratios $P_e/P_d$ and $P_d/P_c$ grow during tidal expansion. The resonance angle carries the identity of the chain, while the slope carries the geometry of the (c, d, e) period-ratio plane. The paper derives $a_{cde}$ analytically from angular-momentum conservation and the invariance of the resonance angle, obtaining $a_{cde}=(4.90 m_b + 4.94 m_c)/(7.27 m_e - m_b)$, and shows by N-body integration that trajectories with $a_{cde}>1.56$ can avoid the destructive (3,5,8) cde resonance and land on the observed ratios. The same machinery identifies a favorable mass model, roughly equal masses with $m_c \approx 1.25 m_b$ and $m_c \approx 1.25 m_e$, and, when masses are unfavorable, requires a time-dependent outward torque on planet b to steepen the effective slope.

What would settle it

Measure the masses of planets b, c, d, and e, for example by radial velocities or transit-timing variations. If the mass ratios put the system in the gray region of the paper's diagram, i.e., $a_{cde} \le 1.56$ with planet e as massive as or heavier than c, the proposed tidal expansion cannot reach the observed period ratios without fine-tuned torques; if planet d turns out to be the lightest planet, the merger origin of d would also not be supported.

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Extended reading notes

Core claim

The paper's central claim is that the Kepler-221 system is the relic of a five-planet resonance chain that was destroyed and partially rebuilt. In the proposed scenario, five planets migrate into a first-order chain of 2:1, 3:2, 4:3, and 3:2 resonances; after the gas disk disperses, an instability (delivered in the simulations as a small kick to one of the two planets d1 and d2) breaks the whole chain, and the two d planets merge into the observed planet d. The (b, c, e) 6:3:1 zeroth-order three-body resonance then reforms only if planets b and c undergo convergent migration, realized either as an outward torque on b (parametrized as an exponentially decaying torque, interpretable as mass loss or dynamical tides) or as anomalously strong tidal damping on c. Once the resonance is re-established, tidal dissipation expands the chain along the (2,3,5) 3BR line toward $P_c/P_b \approx 2.035$ and $P_e/P_c \approx 3.228$. Planet d is not passive in this last phase: if the trajectory crosses the (3,5,8) three-body resonance of c, d, and e, that resonance destabilizes and breaks the (b, c, e) chain in more than 95% of the simulations. A successful run therefore requires the expansion slope $a_{cde} = \Delta(P_e/P_d)/\Delta(P_d/P_c)$ to exceed 1.56, which in turn constrains the planet mass ratios to values close to the peas-in-a-pod model with a slightly more massive planet c.

Load-bearing premise

The load-bearing premise is that, after the collision that created planet d, planets b and c moved toward each other convergently for a while, which the model supplies either as an outward force on planet b or as much stronger tidal braking on planet c; if no such mechanism operated in the real system, the b/c/e resonance never reforms and the whole history fails.

Editorial extensions

If this is right

If the four-phase scenario is correct, planet d is a merger remnant, so its mass should be comparable to or larger than the other planets rather than following a monotonic mass-radius relation.
The requirement $a_{cde}>1.56$ means planets b, c, and e must have nearly equal masses with planet c somewhat heavier; planets c and e would then be low-density super-puff bodies.
The expansion phase carries an age tension: reaching the observed period ratios within roughly 650 Myr needs effective tidal damping $Q_{\mathrm{phy}}<2$, while the reformation phase needs $Q_{\mathrm{phy}}>3$, implying either an older system or an extra dissipation channel such as obliquity tides.
If planet d is initially placed on the wrong side of the (3,5,8) cde resonance, the chain is destroyed in over 95% of simulations, so the present architecture records a narrow avoidance of that resonance.
The same merger-and-reformation logic is offered for Kepler-402 and for the first-order (2,3,4) three-body resonance at the outer edge of K2-138.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper's explicit claims, the slope condition $a_{cde}>1.56$ supplies a generally useful diagnostic: in any compact resonant chain, an interloper planet that is not part of the chain controls whether expansion can proceed, and its observed period ratios encode the relative masses of the chain members.
A clean testable extension would target the predicted mass ordering with radial velocities: if future data place the mass ratios in the gray region of the paper's diagram, e.g., with planet e as massive as or heavier than c, the proposed expansion phase fails without fine-tuned torques, and a merger origin for d becomes much less plausible.
One could also search for analogous systems in the Kepler and TESS samples, pairs of planets straddling a non-resonant third body with the outer pair near a three-body resonance, as candidate products of the same chain-breaking process that need not have started from exactly five planets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a real advance on Goldberg & Batygin (2021). It adds the collision origin of planet d, the post-collision reformation problem, and most usefully the role of d in the expansion phase, where the (c,d,e) (3,5,8) resonance can break the (b,c,e) chain. The analytical slope acde (Eq. C.9) is a clean derivation from angular-momentum and resonance invariants, and the paper is unusually honest about its success rates and assumptions.

What the paper does well: the modular N-body campaign covers each phase separately, quantifies where failures happen, and shows that the zeroth-order 3BR does not reform without convergent b-c migration. The result that d must start on the correct side of the (3,5,8) resonance, and that acde > 1.56 is needed to reach the observed period ratios, is new and testable through future TTV/RV mass measurements. The byproduct discussion of first-order 3BR formation (Appendix A) is a nice bonus for K2-138.

Soft spots, in order. The central claim is feasibility, not uniqueness. The full run needs a prescribed kick, an unmodeled outward torque on b (or Qc/Qb = 0.1), a chosen mass model, and a starting location for d that avoids the dangerous resonance; the paper says so, but each tuned ingredient multiplies the prior. The unmodeled torque on b is a real concern, though the paper does show order-of-magnitude consistency with mass-loss and dynamical-tide estimates; the age tension in Section 6.3 is the one that actually bites. Phase III requires Qphy > 3, the <650 Myr system age requires Qphy < 2, and the successful full simulation corresponds to about 2 Gyr. Invoking obliquity tides is plausible, but it is not implemented, and the required time ordering—weak damping during reformation, strong damping afterward—is exactly the order a Cassini-state obliquity would have to explain rather than assume. Also, the mass constraints are inverse requirements: M2a is chosen partly because it gives acde > 1.56. That is acceptable for a scenario paper, but these are constraints, not predictions. No code or data are released, which makes independent checking harder than it should be.

Who benefits: resonant-chain dynamicists and people working on post-disk instability and giant impacts. The paper deserves a serious referee. I would send it to review and ask the authors either to implement a self-consistent tidal/obliquity model for the age problem or to soften the historical claim to a conditional feasibility study. The analytical slope and the warning about d's role will be citable regardless.

Referee Report

3 major / 5 minor

Summary. The paper proposes a four-phase dynamical history for the four-planet system Kepler-221, in which five planets originally formed in a first-order resonance chain; after disk dispersal an instability breaks the chain, two planets merge to form the present planet d, and tidal dissipation plus a prescribed outward torque on planet b reform and then expand the non-adjacent (b, c, e) 6:3:1 three-body resonance to the observed period ratios. The authors support the scenario with REBOUND/REBOUNDx N-body simulations, a single tuned full-phase simulation, phase-by-phase parameter studies, and an analytical derivation of the expansion slope acde in the (c, d, e) period-ratio plane. The paper derives mass-ratio constraints from the requirement acde > 1.56 and discusses applications to K2-138 and Kepler-402.

Significance. If the scenario is correct, the paper provides a plausible formation channel for non-adjacent three-body resonances with an intervening non-resonant planet, a role for the intermediate planet d in destabilizing or preserving the resonance during tidal expansion, and falsifiable mass-ratio predictions for a system with weak TTVs. Strengths include the clean analytical derivation of the expansion slope in Appendix C, the explicit inclusion of planet d in the expansion dynamics (which previous work neglected), the use of public community codes, and the candid discussion of the model's limitations in Section 6.3 and the conclusions. The mass-ratio and density-ratio constraints in Figure 8 are concrete and testable with radial-velocity follow-up. However, as discussed below, the age versus tidal-damping inconsistency acknowledged by the authors is load-bearing and prevents the paper, in its current form, from establishing that the full four-phase history actually reproduces the Kepler-221 system within its inferred age.

major comments (3)

[Section 6.3, Eq. (14), Conclusion item 6] The paper contains a quantitative internal inconsistency that is load-bearing for the central historical claim. The resonance-reformation simulations require gentle migration with Qphy > 3 (Section 4.4 and Table 3), while the <650 Myr age estimate (Berger et al. 2018) combined with texp ≈ 380 Qphy Myr (Eq. 14) requires Qphy < 2. The fully successful simulation in Figure 2 corresponds to an expansion time of about 2 Gyr, not <650 Myr. The proposed resolution via obliquity tides (Eq. 15) is qualitative and is not implemented in any simulation; moreover, the period of dynamical instability and resonance reformation is exactly when non-zero obliquity would be excited, so the required sequence of weak damping during Phase III and strong damping during Phase IV lacks a demonstrated physical basis. As it stands, the paper has not shown that the four-phase model can reach the observed configuration within the age of the system.
[Section 4.3, Eq. (8), Section 4.4, Figure 3] The reformation of the (b, c, e) zeroth-order 3BR is conditional on a convergent migration between b and c, achieved either by an outward exponentially decaying torque on planet b (Eq. 8) or by strongly enhanced damping on planet c (Qc/Qb = 0.1). The paper parameterizes these mechanisms but does not demonstrate that either operates in Kepler-221; the cited mass-loss and dynamical-tide origins are not modeled from first principles. Since the paper's own simulations show that without such a torque the resonance reforms only as a first-order 3BR or not at all (Figure 3a,d), the physical origin of this torque is a load-bearing assumption. I recommend that the authors either implement a concrete physical mechanism or, if the mechanism is to remain an ansatz, state more prominently that the scenario requires this unmodeled condition and quantify the resulting joint probability of the full sequence rather than only individual phases.
[Section 5.2.2, Figure 8, Eq. (10)-(11)] The mass constraints derived from acde > 1.56 are inverse matching conditions rather than independent predictions. The threshold starts from a specific initial condition, namely the intersection of Pe/Pc = 3 with the (3, 5, 8) resonance line (Appendix C, Figure C.1), and the successful Phase IV simulations use mass model M2a, which was selected after the constraint was derived because it satisfies acde > 1.56. The mass ratios of b, c, and e are therefore constrained only under the assumption that the expansion begins at that particular point and proceeds along the zero-order 3BR line. I ask the authors to clarify explicitly that these are necessary conditions within the assumed scenario, not observational predictions independent of the starting geometry, and to discuss how the constraint changes if the initial Pd/Pc is not exactly at the adopted intersection.

minor comments (5)

[Figure 2 caption and Section 3] The caption for Figure 2 states that the orbital expansion phase is represented by 'panels d and h,' but panel d is already described earlier in the same caption as a 3BR-angle panel; the body text says panel d shows the (b, c, e) period ratios and panel h shows the (c, d, e) period ratios. Please correct the caption to avoid this inconsistency.
[Section 4.3] The sentence 'In panel c and g of Figure 2, the torque applied to planet b amounts to a 2% mass loss in 7.5 Myr' appears to refer to the text of Section 4.3 rather than to anything displayed in Figure 2, whose panels c and g show orbital elements and resonance angles; please clarify the reference.
[Throughout] The manuscript contains numerous spacing artifacts from LaTeX or PDF extraction, including 'e fficient,' 'di fferent,' 'V ogt,' and 'Go´ zdziewski'; these should be cleaned before publication.
[Section 2.3, Table 3] Table 3 is dense and difficult to parse because the parameter column mixes symbols with ranges and notes; consider splitting the table into per-phase blocks or adding a separate parameter-definition table for symbols such as tb_a0,III and tb_Gamma,IV.
[Appendix C, Eq. (C.8)] The sign convention in ΔPe/ΔPc could be made explicit: since Pc decreases while Pe increases during expansion, the raw derivative in Eq. (C.8) is negative, and the positive slope acde in Eq. (C.9) follows after multiplying by -Pc^2/Pd^2; adding one sentence to this effect would prevent reader confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: mass-ratio constraints derive from conservation laws and observed period ratios, and self-cited 3BR results are independently reproduced by the paper's own simulations.

full rationale

The paper's main derived result, acde > 1.56 (Eq. 10/C.9), is obtained analytically from angular-momentum conservation and the invariant 3BR angle, with the observed present-day period ratios used only as the endpoint of the expansion. The planet masses are unmeasured inputs, so the resulting mass/density constraints are genuine inverse-modeling predictions, not parameters fitted to the data they claim to explain. The successful simulation in Fig. 2 uses mass model M2a precisely because it satisfies the analytic condition; the text explicitly calls the outcome 'somewhat tuned and parameter-dependent' and presents it as a feasibility demonstration, not as an independent forecast. The repeated citations to Petit (2021) on zeroth- versus first-order 3BR reformation are corroborated within the paper: over 50 simulations without convergent migration all fail to reform the zeroth-order 3BR, and Appendix A shows the first-order trap directly. The recognized tension between Qphy > 3 needed for Phase III reformation and Qphy < 2 implied by the <650 Myr age is openly stated in Section 6.3 as a limitation with an unmodeled obliquity-tide resolution, which is an assumption gap rather than a circular reduction. No equation is equivalent to its own input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 7 free parameters · 5 assumptions · 1 invented entities

The scenario rests on a chain of postulates: an initial five-planet resonant chain, an instability that breaks it, a merger, a reformation requiring an unmodeled torque, and a tuned expansion trajectory. Each is reasonable within the field but none is independently measured; the mass and tidal parameters are chosen, not derived from data.

free parameters (7)

Planet masses (mb, mc, md, me) = M2a: 5.09, 6.50, 5.09, 5.09 M⊕
Masses are unmeasured (weak TTVs); models M1/M2/M2a/M2d assign values from a mass-radius relation or peas-in-a-pod. The expansion slope acde depends directly on these mass ratios.
Effective tidal quality factor Qphy = 4.6 (fiducial); ranges 3-10 in Phase III
Controls eccentricity damping and expansion speed. Low values speed computation (Qsim=Qphy/100); the age constraint requires Qphy<2, in tension with reformation, so this parameter carries much of the model's uncertainty.
Outward torque on planet b (tb_a0, tb_Γ) = Phase III: tb_a0=1.9e8 yr, tb_Γ=7.5e6 yr; Phase IV: tb_a0=2.1e10 yr, tb_Γ=1.4e9 yr
Imposed via Eq. (8) to force convergent migration in Phase III and to bend the trajectory in Phase IV. The physical source (mass loss or dynamical tides) is cited but not modeled.
Kick magnitude Δa/a on d2 = 0.7% (fiducial; range 0.5-1.2%)
Instantaneous semi-major axis perturbation used to break the resonance chain, standing in for an unspecified instability trigger.
Disk dispersion parameters ta,I, ta,I/te,I, td = ta,I ∈ [1e3,1e6] yr, ratio 150-250, td ∈ [50,1e4] yr
Chosen to leave planets with high post-disk eccentricities (e near 0.01-0.1), which the model needs for a quick merge.
Initial Pd/Pc at start of Phase IV = 1.712-1.728
Starting position of planet d is selected so that the (b,c,e) expansion avoids the (3,5,8) (c,d,e) resonance; departure from this window leads to failure in over 95% of runs.
Mass ratio md2/md1 = 1.02 (range 1-1.04)
Chosen so that the merged planet d's center of mass lands near its observed location.

assumptions (5)

domain assumption Planets formed in a first-order resonance chain via Type-I migration (Sec 2.2, Phase I).
Standard result in planet migration theory; the paper relies on it to set the initial five-planet architecture.
ad hoc to paper The resonance chain breaks after disk dispersal due to an unmodeled instability, represented by a kick on d2 (Sec 4.1).
The trigger is not computed; the paper states 'we do not model the specifics of the resonance breaking process, but simply enforce it'.
ad hoc to paper An outward torque on planet b (or strong damping on c) operates during Phases III-IV, with no confirmed physical origin (Sec 4.3).
Essential for reformation; mass loss and dynamical tides are cited as possible sources but not demonstrated for Kepler-221.
domain assumption Tidal damping follows Eq. (6) and Qsim=Qphy/100 does not alter the evolutionary path (Sec 2.3).
Standard tidal parameterization from Papaloizou et al. (2018); the acceleration is used for computational expediency and asserted not to change outcomes.
domain assumption The analytical slope derivation assumes planet d is stationary during expansion (Appendix C).
Used to derive Eq. (10); Appendix D shows d drifts slightly, which is then treated as a correction.

invented entities (1)

Original planets d1 and d2 independent evidence
purpose: Two pre-merger planets proposed as the origin of the current intermediate planet d; their collision removes the fifth planet and produces the observed non-resonant d.
No direct evidence for these bodies exists; the model predicts d should be dense and possibly massive (merger remnant), which is testable with radial velocity follow-up. That prediction is a falsifiable handle.

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Cite this review

Pith. "Pith review of The Dynamical History of the Kepler-221 Planet System." pith.science (2026). https://pith.science/paper/I5ZXC3XO

@misc{pith2026250201736,
  author       = {Pith},
  title        = {Pith review of: The Dynamical History of the Kepler-221 Planet System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5ZXC3XO}},
  note         = {Machine review of arXiv:2502.01736}
}

read the original abstract

Kepler-221 is a G-type star hosting four planets. In this system, planets b, c, and e are in (or near) a 6:3:1 three-body resonance even though the planets' period ratios show significant departures from exact two-body commensurability. Importantly, the intermediate planet d is not part of the resonance chain. To reach this resonance configuration, we propose a scenario in which there were originally five planets in the system in a chain of first-order resonances. After disk dispersal, the resonance chain became unstable and two planets quickly merged to become the current planet d. In addition, the b/c/e three-body resonance was re-established. We run N-body simulations using REBOUND to investigate the parameter space under which this scenario can operate. We find that our envisioned scenario is possible when certain conditions are met. First, the reformation of the three-body resonance after planet merging requires convergent migration between planets b and c. Second, as has previously pointed out, an efficient damping mechanism must operate to power the expansion of the b/c/e system. We find that planet d plays a crucial role during the orbital expansion phase due to destabilizing encounters of a three-body resonance between c, d, and e. A successful orbital expansion phase puts constraints on the planet properties in the Kepler-221 system including the planet mass ratios and the tidal quality factors for the planets. Our model can also be applied to other planet systems in resonance, such as Kepler-402 and K2-138.

Figures

Figures reproduced from arXiv: 2502.01736 by the authors.

**Figure 1.** Schematic of the formation model for the Kepler-221 planet system investigated in this study. Sequential migration traps five planets in a chain of first-order resonances (panels a+b). After disk dispersal, the resonance chain breaks (panel c) triggering a dynamical instability that results in the merger of planets d1 and d2 (panel d). Convergent migration between planets b and c re-establishes the b, c, and e res… view at source ↗

**Figure 2.** Successful simulation spanning all simulation phases depicted in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Success and failure in reforming the b, c, and e 3BR post-collision. Top panels show the evolution of Pc/Pb and Pe/Pc . The black dashed line and red dashed line correspond to the zeroth-order and first-order 3BR, respectively, and the color of the dots indicates time. The bottom panels show the (b, c, e) 3BR angle. In panels a and d, there is no torque on planet b, and Qphy is the same for all planets. The planets … view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: , while in the other simulations planets c and d1 collide, or some planets are ejected out of the system. The overall success rate in the post-collision phase is around 10% (9 successful in 101 simulations). In the simulation, an outward torque is applied on planet b…

**Figure 5.** Figure 5: Effect of the initial location of planet d on the expansion process of the Kepler-221 system. Two simulations with identical masses of the planet following mass model M1 ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Mechanisms that ensure successful orbital expansion of the Kepler-221 system. Two simulations with identical initial positions of the planets and different tidal strengths, torque on planet b, and planet masses are shown from top to bottom. (i) Top panels: A temporal…

**Figure 7.** Figure 7: Time dependence of torque on b for different physical mechanisms. The dashed lines represent the minimum dynamical torque on (blue) or mass loss rate for (green) planet b required to induce a high enough migration of planet c, which ensures sustained orbital expansio…

**Figure 8.** Figure 8: Constraints on planet mass and density ratios in the orbital expansion phase with the radius of the planets according to [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Mass-radius diagram of the Kepler-221 planets with mass models listed in [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Simulations show an early resonant chain of terrestrial planets, including Theia, can be broken by the giant planet instability and yield a Moon-forming impact, with the present 3.05 Mars-Venus period ratio inherited ...

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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